A new class of stuck unknots in \(\text{Pol}_6\) (Q5948408)

\(\text{Pol}_6\) denotes the set of embedded spatial hexagons in Euclidean 3-space \(E^3\). These are considered as idealized linkages, where the edges of the polygon are rigid bars, the vertices are universal joints and no two edges are allowed to cross each other. Considering continuous motions preserving edge-lengths as isotopies there occurs the question about the number of isotopy classes. For example, if the polygon is planar, then a convex polygon is in the same class and the class only depends on the scheme of edge lengths. Different knot types clearly will lead to different classes. Hence the question to distinguish the different geometric configurations by such classes is more interesting within the knot class. Here the unknotted case is considered. \textit{J. Cantarella} and \textit{H. Johnston} [J. Knot Theory Ramifications 7, No. 8, 1027-1039 (1998; Zbl 0916.57011)] gave an example of a stuck hexagon, i.e., this hexagon is not isotopic to a planar convex one. There are at least three classes with the same scheme of edge lengths in this case, and in the current paper two new classes are found. It is explained that these hexagons are stuck in a stronger way than defined above.